Fully nonlinear logistic equations with sanctuary

TL;DR

This paper investigates the existence, uniqueness, and asymptotic behavior of positive solutions to a fully nonlinear stationary logistic equation with parameter dependence in a bounded domain.

Contribution

It provides a comprehensive analysis of solution existence and uniqueness based on the parameter , and explores the solutions' asymptotic behavior near critical parameter values.

Findings

Existence and uniqueness of solutions depend on the parameter .

Solutions exhibit specific asymptotic behaviors as  approaches boundary points.

The study characterizes the solution structure for the nonlinear logistic equation.

Abstract

For the fully nonlinear stationary logistic equation ${\mathcal F}(x,D^2u)+\mu u=k(x)u^p$ with $p>1$ and $k(x)\geq 0$, in a bounded domain with Dirichlet boundary condition, we determine, in terms of $\mu$, the existence and uniqueness or the nonexistence of a positive solution. Furthermore, we study the asymptotic behavior of the solutions when $\mu$ approaches the boundary points of the existence range.